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What is the factorized form of 𝑥³ + 2𝑥² − 16𝑥 − 32?
What is the factorized form of 𝑥 cubed plus two 𝑥 squared minus 16𝑥 minus 32?
We can factor this using grouping, which means we need to take our polynomial and group the first two terms and the last two terms together once that it’s descending in order. And it already is, and descending order is when you do 𝑥 cubed, then 𝑥 squared, then 𝑥, then the constant.
So looking at our first two terms, the greatest common factor we could take out would be 𝑥 squared. So we would have 𝑥 plus two remaining. Now looking at our next set, we could take out a negative 16, which would leave an 𝑥 plus two remaining.
And this is perfect; we want what’s inside the parentheses to match; they will become a factor. And then the GCFs, the greatest common factors, they will become a factor. So the GCFs make a factor of 𝑥 squared minus 16, and the matching pair make a factor of 𝑥 plus two.
Now the 𝑥 squared minus 16, that’s a difference of squares, and the rule for difference of squares is you square root each of them, and you have 𝑎 plus 𝑏, 𝑎 minus 𝑏. So the square root of 𝑥 squared is 𝑥, and the square root of 16 is four.
So 𝑥 squared minus 16 can be replaced with 𝑥 plus four 𝑥 minus four, and then we bring down the 𝑥 plus two. So this would be our final answer.
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